Coherence Analysis of Financial Analysts’ Recommendations in the Framework of Evidence Theory* Andrew Bronevich, Аlexander Lepskiy, Henry Penikas Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow, 101000, Russia brone@mail.ru, {alex.lepskiy, penikas}@gmail.com Abstract. This article is devoted to the analysis of coherence of financial rec- ommendations with respect to securities of the Russian companies. The study is based on the analysis of approximately 4000 recommendations and forecasts of 23 investment banks with respect to around forty securities of Russian stock market over the period of 2012-2014 years. The predictive history of each of the investment bank was considered as evidence in the framework of evidence theory. The coherence of recommendations was evaluated with the help of the so-called conflict measure between the evidence, which determined on the sub- sets of the set of all evidence. Then the study of coherence was reduced to anal- ysis of values of the conflict measure. This analysis was performed with the help of game-theoretic methods (Shapley index, interaction index), network analysis methods (centralities), fuzzy relation methods, hierarchical clustering methods. Keywords: analysts' recommendations, conflict measure, interaction index, network analysis, hierarchical clustering. 1 Introduction The forecasts and recommendations of financial analysts' (of investment banks) are the important sources of information in decision making by the participants of the financial market. The different aspects connected to the recommendations of financial analysts' are reflected in the research literature. The influence of forecasts of financial analysts' on the investors and the reaction of market on these forecasts is estimated in [13]. The relationship between analysts' fame and the reaction of investors on the corrected forecast is investigated in [2]. The "asymmetry" of analysts’ forecasts and the manipulability of the recommendations is analyzed in [10]. The analysis of the coherence of forecasts and recommendations is one of the im- portant directions of research. The coherence of recommendations is determined as a * The study was implemented in the framework of The Basic Research Program of the Higher School of Economics. This work was supported by the grant 14-07-00189 of RFBR (Rus- sian Foundation for Basic Research). Coherence Analysis of Financial Analysts’ Recommendations 13 rule as similarity of recommendations that is given by different analysts with respect to the same securities. The level of coherence of the recommendations is evaluated more often as an average of all recommendations for a particular security. For exam- ple, the dependence of coherence of the forecasts from a number of the shares charac- teristics was investigated in [7]. In this study, analysis of the coherence of financial analysts' recommendations about the value of the shares of Russian companies in 2012-2014 will be performed in the framework of evidence theory (Dempster-Shafer theory, [4, 14]). Namely, the recommendation of the analyst (the recommendation of the investment bank) is de- scribed as evidence. The evidence determined by the set of focal elements and the mass function. The set of focal elements is a set of intervals of the relative value of the shares corresponding to the recommendations (buy/hold/sell). The mass function is equal to the relative frequency of recommendations in each interval (focal element). In [3] the conflict measure K ∈ [0,1] was introduced on the set of all evidence of this type. This measure characterizes the inconsistency between the evidence. Then the value C = 1 − K has the sense of the degree of the coherence of recommendations. The study, which started in [3], will continue in present article. Namely, the coher- ence of the recommendations will be evaluated and the set of the investment banks will be structured with respect to this coherence. The analysis of coherence will be performed with the help of game-theoretic methods (Shapley index, interaction in- dex), network analysis methods (centralities), fuzzy relation methods, hierarchical clustering methods. In addition, the expressions for some of computational character- istics (Shapley index, interaction index) will be obtained in this study in the terms of the evidence of the type under consideration. The work is structured as follows. The main notions of the evidence theory, the no- tion of the conflict measure are given in Section 2. The axiomatic of the conflict measure is discussed in this section too. The research database is described in Section 3. Section 4 is devoted to the description of evidence corresponding to database and the used conflict measure in the term of evidence. Section 5 is the main part of the work, in which study the coherence by the different methods. Finally, some conclu- sions from research are presented in Section 6. 2 The Evidence Functions Theory and Conflict Measures Let X be a finite set and 2 X be a powerset of X . The mass function and the focal element are the fundamental notions in evidence theory. The mass function is a set function m : 2 X → [0,1] that satisfy the following conditions m(∅) = 0 , ∑ A⊆ X m( A) = 1. (1) The value m( A) characterizes the degree that true alternative from X belongs to the set A ∈ 2 X . The subset A ∈ 2 X is called a focal element if m( A) > 0 . Let A == {A} be a set of all focal elements. Then the pair F = (A , m) is called a body of 14 Andrew Bronevich, Аlexander Lepskiy, Henry Penikas evidence. Let F ( X ) be a set of all bodies of evidence on X . Note that the body of evidence can be considered for an arbitrary nonempty set X , if the set function m : L → [0,1] is defined on the some nonempty set L of subsets from X that satisfy the conditions (1). Let we have two bodies of evidence F1 = (A 1 , m1 ) and F2 = (A 2 , m2 ) . For example, these evidences can be obtained from two sources of information. Then we have a question about the conflict between the two evidences. Historically the function K0 ( F1 , F2 ) connected with Dempster’s combining rule [4, 14] was the first conflict measure: K 0 = K 0 ( F1 , F2 ) = ∑ m1 ( B)m2 (C ) . B ∩C =∅ , B∈A 1 , C∈A 2 The axioms of the conflict measure are considered in [5]. There are few approaches to the estimation of the conflict of evidence. The analyses of these approaches can be found in [3]. It can be allocated conditionally the metric approach [8], the structural approach [11], the algebraic approach [9]. The notion of a conflict measure (and corresponding axioms) was generalized in [3] for arbitrary finite set of evidence. Suppose that we have some finite set of evi- dence M == {F1 ,..., Fl } , Fi ∈F ( X ) , i = 1,..., l . Let 2M be a powerset of M . We shall put by definition that K ( B) = 0 , if B = 1 , B ∈ 2 M and K (∅) = 0 . Note that the conflict measure K 0 that considered on 2 M in the form K0 ({Fi1 ,..., Fik }) = ∑ mi1 ( Ai1 )...mik ( Aik ) , Fis = ({Ais }, mis ) , s = 1,..., k , (2) Ai1 ∩...∩ Aik =∅ satisfies the monotonicity condition: K ( Bʹ) ≤ K ( Bʹʹ) , if B ʹ ⊆ B ʹʹ and B ʹ, B ʹʹ ∈ 2 M . This means that the adding of new evidence to the set of evidence does not reduce the conflict measure. 3 The Description of the Database The conflictness (and coherence as the dual concept of) of the evidences about ana- lysts' forecasts (investment banks) is investigated in this article. The conflictness characterizes in this case the degree of non coherence of forecasts for some set of experts. The study is based on the analysis of approximately 4000 recommendations and forecasts of 23 investment banks with respect to around forty securities of Russian stock market over the period of 2012-2014 years. The databases of the agencies Bloomberg and RBC are the sources of information. The forecasts are presented by experts of the world's largest investment banks including such renowned companies as Goldman Sachs, Credit Suisse, UBS, Deutsche Bank and others. Each investment bank makes recommendations of three types to sell/hold/buy with forecast of target price of the security. The target prices of forecasts are recalculated into the so-called relative values of target prices. The relative value of a target price is Coherence Analysis of Financial Analysts’ Recommendations 15 a ratio of the predicted price to the quotation of the security on the date of the fore- cast. The boundaries of relative prices between the recommendations of various types were determined by maximizing number of recommendations that fall into the "corre- sponding" intervals: [0, 0.97), [0.97, 1.22), [1.22, +∞). Thus, we have nine sets, each of which represents the interval and a label of recommendation type: A1(t ) = [0, 0.97) , A2(t ) = [0.97,1.2) , A3(t ) = [1.2, +∞) , t = 1, 2,3 , where t = 1 ‒ to sell, t = 2 – to hold, t = 3 – to buy. 4 The Description of Evidence and the Used Conflict Measures The belonging of the relative price of the forecast of a certain type (to buy/hold/sell) to one of the three intervals can be considered as an evidence of the investment bank. Then we can found the body of evidence for given investment bank. Let we fixed the i -th investment bank, i = 1,..., l ( l is a number of investment banks), cik( t ) is a num- ber of belonging of relative price to interval Ak( t ) , N i is a general number of forecasts for i -th investment bank. Then mi ( Ak(t ) ) = cik(t ) N i is a frequency of belonging of relative price to interval Ak( t ) . The mass function mi satisfies the normalization condi- tion: ∑ t ∑ k mi ( Ak(t ) ) = 1 for all i = 1,..., l . Then Fi = Ak(t ) , mi ( Ak(t ) ) ( ) is a body of k ,t evidence of i -th investment bank, i = 1,..., l . We can consider that all evidences have the same set of focal elements (even if mi ( Ak(t ) ) = 0 for certain indexes) and all differ- ent focal elements Ak( t ) are pairwise disjoint. Thus, the vector m = (m( s ) )9s =1 , m( k + 3(t −1)) = m( Ak(t ) ) , k = 1, 2,3 , t = 1, 2,3 corresponds bijectively to the body of evi- ( ) . The set of all such evidence forms a simplex dence F = Ak(t ) , m( Ak(t ) ) k ,t 9 S = {m : m ≥ 0 ∀s, ∑ m = 1}. (s) (s) s =1 (s) The formula (2) for calculation of conflict measure K0 ( F1 ,..., Fl ) can be simplified. Proposition 1 [3]. If a bodies of evidence Fi = ({Ak }, mi ( Ak ) ) , i = 1,..., l satisfy the conditions As ∩ Ak = ∅ for s ≠ k , then the conflict measure K0 ( B) , B ⊆ M is equal to K 0 ( B ) = 1 − ∑ k ∏ i: F ∈B mi ( Ak ) . i The following measure K ( B) = 1 − ∑ min mi ( Ak ) , (3) i: Fi ∈B k satisfies also the monotonicity condition and will be considered as a conflict measure below instead of measure K 0 in this paper. 16 Andrew Bronevich, Аlexander Lepskiy, Henry Penikas Let mi( k ) = mi ( Ak ) ∀k , i = 1,..., l and we denote Fi ∈ B ⊆ M for shot i ∈ B . We denote the measure K ( Fi ,..., Fi ) as K (mi ,..., mi ) , if mi ↔ Fi , p = 1,..., s with 1 s 1 s p p consideration of the vector representation of evidence. We will consider a coherence measure C = 1 − K which is defined on 2 M together with a conflict measure K . This measure characterized the degree of coherence of financial analysts' recommendations. Below, we are interested in estimation of increments of the individual analysts' contribution in the total conflict: Δi K ( B) = K ( B ∪{i}) − K ( B) , B ⊆ M \{i} , Δij K ( B) = K ( B ∪ {i, j}) − K ( B ∪{i}) − K ( B ∪{ j}) + K ( B) , B ⊆ M \{i, j} . Let (t ) + = t , t ≥ 0, The following proposition is true for measure (3) and the increments { 0, t < 0. Δi K ( B) and Δ ij K ( B) . Proposition 2. The following equalities are true for any m i , m j ∈ S and B ⊆ S : 1) Kij = K (mi , m j ) = ∑ k max{mi( k ) , m(jk ) } − 1 = 12 ∑ k mi( k ) − m(jk ) ; 2) Δi K (∅) = 0 and Δi K ( B) = ∑ k max {min s∈B ms( k ) , mi( k ) } − 1 = ∑ k ( min s∈B ms( k ) − mi( k ) )+ , if ∅ ≠ B ⊆ M \{i} ; 3) Δ ij K (∅) = Kij and ( ) Δij K ( B) = −∑ k min s∈B ms( k ) − max {mi( k ) , m(jk ) } , if ∅ ≠ B ⊆ M \{i, j} . + Remark 1. The equality 1) in Proposition 2 shows us that the measure of pair con- flict K ( ⋅ , ⋅ ) is a metric on the simplex S . Remark 2. All pair increments of the conflict measure with non empty coalitions are not positive as follows from 3): Δ ij K ( B) ≤ 0 ∀B ≠ ∅ , B ⊆ M \{i, j} . This means that the inclusion of any analyst in the greater coalition increases the conflict measure by a smaller amount than the inclusion of the analyst in the smaller coalition. 5 An Analysis of Evidence Coherence 5.1 The Finding of the Most Conflict Analysts Using the Shapley Vector If the monotone measure K is defined on the set of all subsets of M then we can determine the contribution of i-th analyst in general conflict K (M ) of the set of all analysts M as the difference K (M ) − K (M \{i}) . More accurately the contribution of i-th analyst in general conflict can be determined as a average contribution in the conflict of the group (coalition) of analysts B : Δi K ( B) = K ( B ∪{i}) − K ( B) , where the average is computed for all groups (coalitions) of analysts B , B ⊆ M \{i} . In this case we will get so called Shapley value [15], which is widely used in the coalition Coherence Analysis of Financial Analysts’ Recommendations 17 (cooperative) game theory: vi = ∑ B⊆ M \{i}αl ( B ,1) Δi K ( B) , i = 1,..., l , l − s − r !s ! α l ( s, r ) = ( (l − r +1))! , s + r = 1,..., l . The vector v = (vi )il =1 is called by Shapley vector l and it satisfies the condition ∑ i =1 vi = K (M ) . We will find an expression for the Shapley values of conflict measure (3) in terms of evidence Fi ↔ mi , i = 1,..., l . Proposition 3. The following formula is true for Shapley values of conflict meas- { ure (3): vi = ∑ ∅≠ B ⊆ M \{i}αl ( B ,1) ∑ k max mk(i ) , min mk( s ) − l −l 1 , i = 1,..., l . s∈B } The contributions of all investment banks in the general conflictness of recommen- dations in period 2012-2014 are shown in the Fig. 1. These contributions were esti- mated with the help of Shapley values. The general conflictness for all 23 investment banks is equal 0.625 . 0.1 0.08 Shapley value 0.06 0.04 0.02 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 investment banks Fig. 1. The Shapley values of investment banks Remark 3. The following denotations of investment banks are used on Fig. 1–4, in Tables 1– 2: 1 ‒ Alfa-Bank, 2 ‒ Aton Bank, 3 ‒ BCS, 4 ‒ Veles Capital, 5 ‒ VTB Capital, 6 ‒ Gazprom- bank, 7 ‒ Metropol Bank, 8 ‒ Discovery Bank, 9 ‒ Renaissance Capital, 10 ‒ Uralsib Capital, 11 ‒ Finam, 12 ‒ Barclays, 13 ‒ Citi group, 14 ‒ Credit suisse, 15 ‒ Deutsche Bank, 16 ‒ Goldman Sachs, 17 ‒ HSBC, 18 ‒ J.P. Morgan, 19 ‒ Morgan Stanley, 20 ‒ Raiffeisen, 21 ‒ Rye. Man&GorSecurities, 22 ‒ Sberbank CIB, 23 ‒ UBS. The interrelation between the Shapley values of investment banks and the profita- bility of forecasts was analyzed in [3]. 5.2 An Analysis of the Mutual Coherence of the Recommendations of Analysts with the help of Interaction Index In addition to the detection of key analysts (investment banks) with the help of Shap- ley values that have the greatest influence on the coherence of forecasts, it is im- portant to analyze the mutual influence of investment banks on the coherence of fore- casts. This can be done with the help of the so-called interaction index [6], which is equal I (T ) = ∑ B ⊆ M \T α l ( B , T ) ∑ C ⊆T ( −1) T \ C K ( C ∪ B ) for arbitrary coalition T and monotone measure K , defined on the finite set M , M = l . The interaction index I (T ) of the set of analysts T characterizes in our case the value of added con- 18 Andrew Bronevich, Аlexander Lepskiy, Henry Penikas tribution (synergistic effect) of this set in general conflict as compared with the sum- mary contribution of separate analysts and improper subsets of T in the conflict. In particular, I ({i}) = vi is a Shapley value, i = 1,..., l . The interaction index for coali- tions from two elements I ({i, j}) = I ij has an important value. This index was intro- duced earlier in [12]: Iij = ∑ B⊆ M \{i , j}αl ( B , 2) Δij K ( B) . The interaction index has value in the interval [−1,1] . If I ij is close to 1, then this means that these analysts in pair increase the conflict in combination with the other coalitions to a larger value than each of them individually. If I ij is close to −1 , then the union of two analysts in the pair will not cause the synergistic effect in calculation of conflict. We will find an expression for the pair interaction index of the conflict measure (3) in terms of evi- dence Fi ↔ mi , i = 1,..., l . Proposition 4. The following formula is true for pair interaction index of the con- flict measure (3) I ij = l −11 Kij − ∑ ∅≠ B ⊆ M \{i , j } k ( ) α l ( B , 2 ) ∑ min s∈B ms( k ) − max {mi( k ) , m(jk ) } . + The values of the interaction index I ij that characterized the contributions of pairs of investment banks in the general conflict of forecasts about the value of shares of Russian companies in period 2012-2014 are shown in Table 1. The values for which I ij ≥ 0.013 are indicated only in the table. Table 1. The values of the interaction index I ij , I ij ≥ 0.013 11 14 19 20 21 22 23 6 -0,017 0,013 7 0,013 -0,02 11 -0,015 12 -0,023 -0,013 -0,015 0,013 13 -0,015 14 -0,014 0,014 0,015 16 -0,014 Since we are interested in the coherence measure of recommendations С = 1 − K then and I ij (C ) = − I ij ( K ) , then the pair with negative and large absolute values are interesting for us in Table 1. It is the pairs (in decreasing order of I ij ): (12,14), (7,21), (6,11), (11,21), (12,20), (13,23). 5.3 A Network Analysis of the Coherence of Analysts' Recommendations We consider the coherence graph of recommendations G = ( N , C) on the set of all investment banks, where N = {ni } be a set of all nodes (investment banks), C = {Сij } be a set of edges with weights Сij = 1 − K ij and K ij be a value of conflict measure between the i -th and j -th investment banks, which calculated by formula (3). We Coherence Analysis of Financial Analysts’ Recommendations 19 can consider the “roughenned” coherence graph instead of the graph G for a better ⎧1, K ij < h, visualization with weights Сij = ⎨ where h is a threshold value. The such ⎩0, K ij ≥ h, graph, which constructed by the data of value of shares of Russian companies in peri- od 2012-2014, is shown in the Fig. 2 for h = 0.15 . Fig. 2. The coherence graph of recommendations of investment banks The matrix of pair coherence of recommendations C = {Сij } is a symmetric and non-negative. We investigate the problem of finding such investment banks, which have a most influence on coherence of recommendations. We will consider the so- called eigenvector centrality [1]. This centrality takes into account not only neighbor links but also distant links of nodes. The calculation of the measure of centrality for each node associated with the solution of the eigenvector problem with respect to the adjacency matrix A of the network graph. The vector of the relative centralities x is an eigenvector of the adjacency matrix corresponding to the largest eigenvalue λmax , i.e. Ax = λmax x . centrality value 0.23 0.22 0.21 0.2 0.19 0.18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 investment banks Fig. 3. The values of coordinates of centrality vector for the coherence graph of recommenda- tions of investment banks We have λmax = 17.9 for the data of value of shares of Russian companies in peri- od 2012-2014. The values of coordinates of corresponding eigenvector (centrality vector) are shown in the Fig. 3. As can be seen from this figure, the greatest influence on the coherence of the recommendations in accordance with the values coordinates of the centrality vector have the banks (in descending order of influence) 9,1,18,15,23, etc. 20 Andrew Bronevich, Аlexander Lepskiy, Henry Penikas The centrality vector correlated greatly and negatively with the Shapley vector. The corresponding correlation coefficient is equal to −0.86 . However, pairwise coherencies of recommendations do not give a complete picture of the more complex (not pairwise) interactions. This kind of interaction can be re- vealed with the help of analysis of the cluster structures of relations on the set of evi- dence, which is given by a conflict measure. 5.4 An Analysis of Fuzzy Relations on the Set of Evidence Let M == {F1 ,..., Fl } be a set of evidence. Then the pair conflict measure Kij = K ( Fi , Fj ) and the corresponding coherence measure Сij = 1 − K ij can be con- sidered as binary fuzzy relations, which are given on the Cartesian square M 2 . The relation C = (Cij ) is a similarity relation (i.e. reflexive and symmetric fuzzy relation) [18]. It is easy to verify that the relation C = (Cij ) is not a max-min transitive relation [18]. But we can construct the relation Cˆ = (Cˆij ) with the help of a transitive closure operator Cˆ = U ∞ n =1 C n . This relation will be a max-min transitive relation and, conse- quently, will be a fuzzy equivalence relation. Then the relation Kˆ = 1 − Cˆ will be dissimilitude relation. The dissimilitude relation K̂ defines the ultrametric on M 2 (i.e. K̂ satisfies the axioms: 1) Kˆ ( F , G) = 0 ⇔ F = G ; 2) Kˆ (F , G) = Kˆ (G, F ) ; 3) Kˆ (F , G) ≤ max{Kˆ (F , J ), Kˆ ( J , G)} for all F , G, J ∈ M ). Thus, the matrix ( Kˆ ij ) can considered as a matrix of distances between the ana- lysts. The corresponding matrix of coherence Cˆ = (Cˆij ) can considered as a similarity matrix between the investment banks. The structure of coherence of investment bank recommendations can be analyzed with the help of the α-cut Cˆα = {( F , G) : Cˆ ( F , G) ≥ α} , α ∈ (0,1] of the fuzzy similar- ity relation Ĉ . For every fixed α ∈ (0,1] the set Ĉα defines the equivalence relation, which induces a partition of evidence M on the equivalence classes. The equivalence classes of coherence indicated in Table 2 (only not singletons) for some values of α ∈ (0,1] for the data of value of shares of Russian companies in peri- od 2012-2014. Each of these classes represents set of analysts, whose recommenda- tions have а large degree of coherence. This degree is defined by threshold α. Table 2. The equivalence classes of coherence of investment bank recommendations α equivalence classes 0.95 (3,17), (7,21) 0.9 (3,5,17), (1,9,16,22), (8,15), (7,21) 0,85 (1,…,11,13,15,…23) Coherence Analysis of Financial Analysts’ Recommendations 21 5.5 A Cluster Analysis of the Coherence of Analysts' Recommendations We consider the matrix Kˆ = 1 − Cˆ , where Ĉ is a transitive closure of similarity rela- tion C = 1 − K , K = ( K ij ) and K ij is a value of conflict measure between the i -th and j -th investment banks, which calculated by formula (4). A conflict measure considered on the set of evidence M == {F1 ,..., Fl } . The cluster analysis of coherence of analysts' recommendations will be performed using one of the methods of hierarchical clustering. For example, we will use the Unweighted Pair-Group Method Using Arithmetic Averages (UPGMA) [16], which is the most simple and popular from the agglomerative methods of clustering. In this method a union of closest clusters is performed on each iteration step beginning with the singletons (clusters with the unit cardinality). The binary tree of decision (or den- drogram) is constructed as a result of the algorithm. The ultrametricity of data guaran- tees the uniqueness of construction of such tree [17]. The dendrogram of coherence of investment bank recommendations for the data of value of shares of Russian compa- nies in period 2012-2014 is shown in the Fig. 41. The dendrogram presents the full picture of the cluster structures. In particular, we can indicate the following basic cluster structures of investment banks with respect to the coherence of recommenda- tions (these clusters highlighted in various shades of gray in the Fig. 2): (((7,21), 11), ((3,17), 5)), (((1,9), (16,22)), (13,23)), ((8,15), 10). We can see that the result of hier- archical clustering agrees well with the partition of similarity relation Ĉ on the equivalence classes. Fig. 4. The dendrogram of cluster structure of coherence of investment bank recommendations 1 The dendrogram was obtained with the help of the utility http://genomes.urv.cat/UPGMA/ 22 Andrew Bronevich, Аlexander Lepskiy, Henry Penikas 6 Conclusion In this paper, the coherence of investment bank recommendations was studied for the data of value of shares of Russian companies in period 2012-2014. The specific of the study consists in using the conflict measure defined in the framework of the belief function theory for determination of the coherence of recommendations. The analysis of coherence was reduced to analysis of values of the conflict measure. This analysis was performed with the help of game-theoretic methods (Shapley index, interaction index), network analysis methods (centralities), fuzzy relation methods, hierarchical clustering methods. The following results were obtained: ─ the ranking of investment banks with respect to their contribution to the overall coherence of the recommendations using the Shapley value was obtained; ─ the contributions of the separate pairs of investment banks in the total conflict of recommendations of the Russian companies with the help of the interaction index were evaluated; ─ the investment banks rendering the greatest influence on the coherence of the rec- ommendations were detected with the help of the analysis of the centrality; ─ the sets of analysts whose recommendations have a greater degree of coherence were identified with the help of analysis of fuzzy similarity relations generated by the coherence measure; ─ the main cluster structures of investment banks with respect to coherence of the recommendations were identified by the method of hierarchical clustering; ─ the expressions for some of the calculated parameters (Shapley values, interaction index) were obtained in the terms of evidence. 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